Asymptotic behaviour of the predictive density in the exchangeable case

Abstract

Consider the exchangeable case, in which ( x 1, …, x n ) is an observation of the n-variate random variable ( X 1, …, X n ) with density ∫ Θ П n i=1f(x i¦θ) dQ(θ) , where f(x¦θ) is a known model and Q is an unknown mixing distribution. The usual predictive density obtained from a prior p( θ) and the observation ( x 1, …, x n ) is considered as a possible estimator for the one-variate marginal density f Q(y) = ∫ Θ f(y¦θ) dQ(θ) and its asymptotic behaviour, under P Q (the underlying probability model for X 1, …, X n , …), is studied. We show, under mild conditions, that the predictive density is an asymptotically unbiased estimator of f Q ( y). We also prove that it is consistent in mean squared error if and only if Q is a degenerate distribution. Finally, we consider a pooled version of this estimator and prove its L 1-consistency (which allows to consider it as a global density estimator) and asymptotic normality.